Enrichment Through Variation

R. Gordon and A. J. Power

Abstract: We show that, for a closed bicategory W,
the 2-category of tensored W-categories and all
W-functors between them is equivalent to the 2-category of
closed W- representations and maps of such, which in turn is
isomorphic to a full sub-2-category of Lax(W, Cat). We
further show that, if w is a locally dense subbicategory of
W and W is biclosed, then the 2-category of
W-categories having tensors with 1-cells of w embeds
fully into the 2-category of w-representations. This allows
us to generalize Gabriel-Ulmer duality to W-categories and
to prove, for W-categories, that for locally finitely
presentable A and for B admitting finite tensors and
filtered colimits, the category of W-functors from Af
to B is equivalent to that of finitary W-functors
from A to B.